-
question_answer1)
Let \[A=\{p,q,r\}.\] which of the following is an equivalence relation in A?
A)
\[{{R}_{1}}=\{(p,q),(q,r),(p,r),(p,q)\}\] done
clear
B)
\[{{R}_{2}}=\{(r,q),(r,p),(r,r),(q,q)\}\] done
clear
C)
\[{{R}_{3}}=\{(p,p),(q,q),(r,r),(p,q)\}\] done
clear
D)
None of these done
clear
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question_answer2)
Let \[R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}\] be a relation on the set \[A=\{1,2,3,4\}...\] The relation R is
A)
Reflexive done
clear
B)
Transitive done
clear
C)
Not symmetric done
clear
D)
A function done
clear
View Solution play_arrow
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question_answer3)
Let \[P=\{(x,y):\left| {{x}^{2}}+{{y}^{2}} \right|=1,x,y\in R\}.\] Then P is
A)
Reflexive done
clear
B)
Symmetric done
clear
C)
Transitive done
clear
D)
Anti-symmetric done
clear
View Solution play_arrow
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question_answer4)
Let \[\rho \] be the relation on the set R of all real numbers defined by setting \[a\rho b\] if f\[\left| a-b \right|\le \frac{1}{2}.\]then, \[\rho \] is
A)
Reflexive and symmetric but not transitive done
clear
B)
Symmetric and transitive but not reflexive done
clear
C)
Transitive but neither reflexive nor symmetric done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer5)
The domain of the function f(x)=\[\frac{1}{\sqrt{^{10}{{C}_{x-1}}-3{{\times }^{10}}{{C}_{x}}}}\] contains the points
A)
\[9,10,11\] done
clear
B)
\[9,10,12\] done
clear
C)
All natural numbers done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer6)
Let \[f(x)={{x}^{2}}+3x-3,x>0.\] If n points \[{{x}_{1}},{{x}_{2}},{{x}_{3}},...{{x}_{n}}\] are so chosen on the x-axis such that (i) \[\frac{1}{n}\sum\limits_{i=1}^{n}{{{f}^{-1}}({{x}_{i}})}=f\left( \frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}} \right)\] (ii) \[\sum\limits_{i=1}^{n}{{{f}^{-1}}}({{x}_{i}})=\sum\limits_{i=1}^{n}{{{x}_{i}}},\] where \[{{f}^{-1}}\] denotes the inverse of f. The value of \[\frac{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}{n}=\]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer7)
\[f(x)=\left| x-1 \right|,f:{{R}^{+}}\to R\] and \[g(x)={{e}^{x}},\] \[g:[(-1,\infty )\to R].\] If the function fog (x) is defined, then its domain and range respectively are
A)
\[(0,\infty )\,\,and\,\,[0,\infty )\] done
clear
B)
\[[-1,\infty )\,\,and\,\,[0,\infty )\] done
clear
C)
\[[-1,\infty )and\left[ 1-\frac{1}{e},\infty \right)\] done
clear
D)
\[[-1,\infty )and\left[ \frac{1}{e}-1,\infty \right)\] done
clear
View Solution play_arrow
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question_answer8)
Let \[f:[4,\infty )\to [1,\infty )\]be a function defined by \[f(x)={{5}^{x(x-4)}},\] then \[{{f}^{-1}}(x)\]is
A)
\[2-\sqrt{4+{{\log }_{5}}x}\] done
clear
B)
\[2+\sqrt{4+{{\log }_{5}}x}\] done
clear
C)
\[{{\left( \frac{1}{5} \right)}^{x(x-4)}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer9)
Let f and g be functions from R To R defined as \[f(x)=\left\{ \begin{matrix} 7{{x}^{2}}+x-8,x\le 1 \\ 4x+5,1<x\le 7 \\ 8x+3,x>7 \\ \end{matrix},g(x)=\left\{ \begin{matrix} \left| x \right|,x<-3 \\ 0,-3\le x<2 \\ {{x}^{2}}+4,x\ge 2 \\ \end{matrix} \right. \right.\] Then
A)
\[(fog)(-3)=8\] done
clear
B)
\[(fog)(9)=683\] done
clear
C)
\[(gof)(0)=-8\] done
clear
D)
\[(gof)(6)=427\] done
clear
View Solution play_arrow
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question_answer10)
Let \[f:R\to R\] be given by \[f(x)={{(x+1)}^{2}}-1,x\ge -1.\] Then, \[{{f}^{-1}}(x),\] is
A)
\[-1+\sqrt{x+1}\] done
clear
B)
\[-1-\sqrt{x+1}\] done
clear
C)
Does not exist because f is not one-one done
clear
D)
Does not exist because f is not onto done
clear
View Solution play_arrow
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question_answer11)
Let R be a relation over the \[N\times N\] and it is defined by (a, b) R (c, d) \[\Rightarrow a+d=b+c.\] Then R is
A)
Reflexive only done
clear
B)
Symmetric only done
clear
C)
Transitive only done
clear
D)
An equivalence relation done
clear
View Solution play_arrow
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question_answer12)
R is a relation from \[\{11,12,13\}\] to \[\{8,10,12\}\]defined by \[y=x-3.\] The relation \[{{R}^{-1}}\] is
A)
\[\{(11,8),(13,10)\}\] done
clear
B)
\[\{(8,11),(10,13)\}\] done
clear
C)
\[\{(8,11),(9,12),(10,13)\}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
Let \[R=\{(x,y):x,y\in N\] and \[{{x}^{2}}-4xy+3{{y}^{2}}=0\},\] Where N is the set of all natural numbers. Then the relation R is:
A)
Reflexive but neither symmetric nor transitive. done
clear
B)
Symmetric and transitive. done
clear
C)
Reflexive and symmetric. done
clear
D)
Reflexive and transitive. done
clear
View Solution play_arrow
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question_answer14)
Let n be a fixed positive integer. Define a relation R in the set Z of integers by aRb if and only if \[\frac{n}{a-b}.\]The relation R is
A)
reflexive done
clear
B)
Symmetric done
clear
C)
Transitive done
clear
D)
An equivalence relation done
clear
View Solution play_arrow
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question_answer15)
Let \[n(A)=4\] and \[n(B)=6.\] The number of one to one functions from A to B is
A)
24 done
clear
B)
60 done
clear
C)
120 done
clear
D)
360 done
clear
View Solution play_arrow
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question_answer16)
The relation R defined in \[A=\{1,2,3\}\] by aRb, if \[\left| {{a}^{2}}-{{b}^{2}} \right|\le 5.\] Which of the following is false?
A)
\[R=\{(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(3,2)\}\] done
clear
B)
\[{{R}^{-1}}=R\] done
clear
C)
Domain of \[R=\{1,2,3\}\] done
clear
D)
Range of \[R=\{5\}\] done
clear
View Solution play_arrow
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question_answer17)
Total number of equivalence relations defined in the set \[S=\{a,b,c\}\] is:
A)
5 done
clear
B)
3! done
clear
C)
\[{{2}^{3}}\] done
clear
D)
\[{{3}^{3}}\] done
clear
View Solution play_arrow
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question_answer18)
Let \[f(x)=\frac{x}{1+{{x}^{2}}}\] and \[g(x)=\frac{{{e}^{-x}}}{1+[x],}\], where \[[x]\]is the greatest integer less than or equal to x, then
A)
\[D(f+g)=R-[-2,0)\] done
clear
B)
\[D(f+g)=R-[-1,0)\] done
clear
C)
\[R(f)\cap R(g)=\left[ -2,\frac{1}{2} \right]\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
Let \[f:R\to R\] be a function defined by \[f(x)=\frac{x-m}{x-n},\] where \[m\ne n,\] then
A)
f is one-one onto done
clear
B)
f is one-one into done
clear
C)
f is many-one onto done
clear
D)
f is many-one into done
clear
View Solution play_arrow
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question_answer20)
Let A and B be two finite sets having m and n elements respectively. Then, the total number of mapping from A and B is:
A)
\[mn\] done
clear
B)
\[{{2}^{mn}}\] done
clear
C)
\[{{m}^{n}}\] done
clear
D)
\[{{n}^{m}}\] done
clear
View Solution play_arrow
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question_answer21)
The domain of the function\[f(x){{=}^{24-x}}{{C}_{3x-1}}{{+}^{40-6x}}{{C}_{8x-10}}\] is,
A)
\[\{2,3\}\] done
clear
B)
\[\{1,2,3\}\] done
clear
C)
\[\{1,2,3,4\}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer22)
If \[f(x)=5lo{{g}_{5}}x\] then \[{{f}^{-1}}(\alpha -\beta )\] where \[\alpha ,\beta \in R\]is equal to
A)
\[{{f}^{-1}}(\alpha )-{{f}^{-1}}(\beta )\] done
clear
B)
\[\frac{{{f}^{-1}}(\alpha )}{{{f}^{-1}}(\beta )}\] done
clear
C)
\[\frac{1}{f(\alpha -\beta )}\] done
clear
D)
\[\frac{1}{f(\alpha )-f(\beta )}\] done
clear
View Solution play_arrow
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question_answer23)
Let a relation R in the set R of real numbers be defined as (a, b) \[\in R\] if and only if \[1+ab>o\] for all \[a,b\in R\]. The relation R is
A)
Reflexive and symmetric done
clear
B)
Symmetric and transitive done
clear
C)
Only transitive done
clear
D)
An equivalence relation done
clear
View Solution play_arrow
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question_answer24)
The number of linear functions f satisfying \[f(x+f(x))=x+f(x)\forall x\in R\] is
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
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question_answer25)
Let \[f:R\to R\] be function defined by \[f(x)=sin(2x-3),\] then f is
A)
Injective done
clear
B)
surjective done
clear
C)
bijective done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
Let \[f:\{x,y,z\}\to \{1,2,3\}\] be a one-one mapping such that only one of the following three statements is true and remaining two are false: \[f(x)\ne 2,f(y)=2,f(z)\ne 1\], then
A)
\[f(x)>f(y)>f(z)\] done
clear
B)
\[f(x)<f(y)<f(z)\] done
clear
C)
\[f(y)<f(x)<f(z)\] done
clear
D)
\[f(y)<f(z)<f(x)\] done
clear
View Solution play_arrow
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question_answer27)
If \[f:R\to R,f(x)=\left\{ \begin{matrix} x\left| x \right|-4,x\in Q \\ x\left| x \right|-\sqrt{3}\,x\notin Q \\ \end{matrix}, \right.\] then \[f(x)\] is
A)
one to one and onto done
clear
B)
Many to one and onto done
clear
C)
one to one and into done
clear
D)
Many to one and into done
clear
View Solution play_arrow
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question_answer28)
If \[f:R\to S,\]defined by \[f(x)=sinx-\sqrt{3}\cos x+1,\] is onto, then the interval of S is
A)
\[[-1,3]\] done
clear
B)
\[[-1,1]\] done
clear
C)
\[[0,1]\] done
clear
D)
\[[0,3]\] done
clear
View Solution play_arrow
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question_answer29)
Let \[A=\{1,2,3\}\] and \[B=\{a,b,c\}.\] If f is a function from A to B and g is a one-one function from A to B, then the maximum number of definitions of
A)
f is 9 done
clear
B)
g is 9 done
clear
C)
f is 27 done
clear
D)
g is 16 done
clear
View Solution play_arrow
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question_answer30)
Let \[f(x)=sinx\] and \[g(x)=lo{{g}_{e}}\left| x \right|.\] If the ranges of the composition functions fog and gof are \[{{R}_{1}}\] and \[{{R}_{2}}\], respectively, then
A)
\[{{R}_{1}}=\{u:-1\le u<1\},{{R}_{2}}=\{v:-\infty <v<0\}\] done
clear
B)
\[{{R}_{1}}=\{u:-\infty <u<0\},{{R}_{2}}=\{v:-\infty <v<0\}\] done
clear
C)
\[{{R}_{1}}=\{u:-1<u<1\},{{R}_{2}}=\{v:-\infty <v<0\}\] done
clear
D)
\[{{R}_{1}}=\{u:-1\le u\le 1\},{{R}_{2}}=\{v:-\infty <v\le 0\}\] done
clear
View Solution play_arrow
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question_answer31)
Let R be a relation on \[N\times N\] defined by \[(a,b)R(c,d)\Rightarrow ad(b+c)=bc(a+d).R\] is
A)
A partial order relation done
clear
B)
An equivalence relation done
clear
C)
An identity relation done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer32)
If \[f:R\to R\] and \[g:R\to R\] are given by \[f(x)=\left| x \right|\] and \[g(x)=[x]\] for each \[x\in R,\] then \[[x\in R:g(f(x))\le f(g(x))\}=\]
A)
\[Z\cup (-\infty ,0)\] done
clear
B)
\[(-\infty ,0)\] done
clear
C)
\[Z\] done
clear
D)
R done
clear
View Solution play_arrow
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question_answer33)
The number of surjection from \[A=\{1,2,...,n\},n\ge 2ontoB=\{a,b\}\] is
A)
\[^{n}{{P}_{2}}\] done
clear
B)
\[{{2}^{n}}-2\] done
clear
C)
\[{{2}^{n}}-1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
The range of the function \[f(x){{=}^{7-x}}{{P}_{x-3}}\] is
A)
\[\{1,2,3\}\] done
clear
B)
\[\{1,2,3,4,5,6\}\] done
clear
C)
\[\{1,2,3,4\}\] done
clear
D)
\[\{1,2,3,4,5\}\] done
clear
View Solution play_arrow
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question_answer35)
Inverse of the function \[f:R\to (-\infty ,1)\]given by\[f(x)=1-{{2}^{-x}}\] is
A)
\[-{{\log }_{2}}(1-x)\] done
clear
B)
\[-{{\log }_{2}}(x)\] done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer36)
Let \[A=R-\{3\},B=R-\{1\},\] and let \[f:A\to B\] be defined by \[f(x)=\frac{x-2}{x-3}f\] is
A)
Not one-one done
clear
B)
Not onto done
clear
C)
Many-one and onto done
clear
D)
One-one and onto done
clear
View Solution play_arrow
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question_answer37)
The image of the interval [1, 3] under the mapping \[f:R\to R,\] given by \[f(x)=2{{x}^{3}}-24x+107\] is
A)
\[[0,89]\] done
clear
B)
\[[75,89]\] done
clear
C)
\[[0,75]\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer38)
If \[f:R\to R\] is given by \[f(x)=\frac{{{x}^{2}}-4}{{{x}^{2}}+1},\] then the function f is
A)
many-one onto done
clear
B)
many-one into done
clear
C)
one-one into done
clear
D)
one-one onto done
clear
View Solution play_arrow
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question_answer39)
Which of the following function is (are) injective map (s)?
A)
\[f(x)={{x}^{2}}+2,x\in (-\infty ,\infty )\] done
clear
B)
\[f(x)=\left| x+2 \right|,x\in [-2,\infty )\] done
clear
C)
\[f(x)=(x-4)(x-5),x\in (-\infty ,\infty )\] done
clear
D)
\[f(x)=\frac{4{{x}^{2}}+3x-5}{4+3x-5{{x}^{2}}},\,\,x\in (-\infty ,\infty )\] done
clear
View Solution play_arrow
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question_answer40)
\[f:R\to \]defined by \[f(x)=(x-1)(x-2)(x-3)\] is
A)
one-one and into done
clear
B)
one-one and onto done
clear
C)
Many one and into done
clear
D)
many-one and onto done
clear
View Solution play_arrow
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question_answer41)
The inverse of \[f(x)=\frac{2}{3}\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}\] is
A)
\[\frac{1}{3}{{\log }_{10}}\frac{1+x}{1-x}\] done
clear
B)
\[\frac{1}{2}{{\log }_{10}}\frac{2+3x}{2-3x}\] done
clear
C)
\[\frac{1}{3}{{\log }_{10}}\frac{2+3x}{2-3x}\] done
clear
D)
\[\frac{1}{6}{{\log }_{10}}\frac{2-3x}{2+3x}\] done
clear
View Solution play_arrow
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question_answer42)
If \[g(f(x))=\left| \sin x \right|\] and \[f(g(x))={{(sin\sqrt{x})}^{2}},\]then
A)
\[f(x)=si{{n}^{2}}x,g(x)=\sqrt{x}\] done
clear
B)
\[f(x)=sinx,g(x)=\left| x \right|\] done
clear
C)
\[f(x)={{x}^{2}},g(x)=sin\sqrt{x}\] done
clear
D)
f and g cannot be determined. done
clear
View Solution play_arrow
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question_answer43)
If \[f(x)=\frac{x}{x-1},\] then \[\frac{(fofo...of)(x)}{19times}\] is equal to:
A)
\[\frac{x}{x-1}\] done
clear
B)
\[{{\left( \frac{x}{x-1} \right)}^{19}}\] done
clear
C)
\[\frac{19x}{x-1}\] done
clear
D)
x done
clear
View Solution play_arrow
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question_answer44)
The graph of the function \[\cos x\,\cos \,(x+2)-co{{s}^{2}}(x+1)\] is
A)
A straight line passing through \[(0,-si{{n}^{2}}1)\] with slope 2 done
clear
B)
A straight line passing through (0, 0) done
clear
C)
A parabola with vertex \[(1,-si{{n}^{2}}1)\] done
clear
D)
A straight line passing through the point \[\left( \frac{\pi }{2},-{{\sin }^{2}}1 \right)\]and parallel to the x-axis. done
clear
View Solution play_arrow
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question_answer45)
If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}\] (gof) \[(x)=2{{x}^{2}}-5x+2,\] then \[f(x)\] is equal to
A)
\[2x-3\] done
clear
B)
\[2x+3\] done
clear
C)
\[2{{x}^{2}}+3x+1\] done
clear
D)
\[2{{x}^{2}}-3x-1\] done
clear
View Solution play_arrow
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question_answer46)
The number of objective functions from a set A to itself when A contains 106 elements, is
A)
106 done
clear
B)
\[{{(106)}^{2}}\] done
clear
C)
\[(106)!\] done
clear
D)
\[{{2}^{106}}\] done
clear
View Solution play_arrow
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question_answer47)
If \[f(x)=\frac{\sin ([x]\pi )}{{{x}^{2}}+x+1}\] where [.] denotes the greatest integer function, then
A)
f is one-one done
clear
B)
f is not one-one and non-constant done
clear
C)
f is a constant function done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer48)
Let \[f(x)=2{{x}^{2}},g(x)=3x+2\] and \[fog(x)=18{{x}^{2}}+24x+c,\]Then c=
A)
2 done
clear
B)
8 done
clear
C)
6 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer49)
If \[f(x)=x-{{x}^{2}}+{{x}^{3}}-{{x}^{4}}+...to\] \[\infty \] for \[\left| x \right|<1,\] then \[{{f}^{-1}}(x)=\]
A)
\[\frac{x}{1+x}\] done
clear
B)
\[\frac{x}{1-x}\] done
clear
C)
\[\frac{1-x}{x}\] done
clear
D)
\[\frac{1}{x}\] done
clear
View Solution play_arrow
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question_answer50)
If \[f(x)=ax+b\] and \[g(x)=cx+d,\] then\[f\{f\{x\}=g\{f(x)\}\] is equivalent to?
A)
\[f(a)=g(c)\] done
clear
B)
\[f(b)=g(b)\] done
clear
C)
\[f(d)=g(b)\] done
clear
D)
\[f(c)=g(a)\] done
clear
View Solution play_arrow
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question_answer51)
Let \[f(z)=sinz\] and \[g(z)=cosz.\] If * denotes a composition of functions, then the value of \[{{(f+ig)}^{*}}(f-ig)\] is:
A)
\[i{{e}^{-{{e}^{-iz}}}}\] done
clear
B)
\[i{{e}^{-{{e}^{iz}}}}\] done
clear
C)
\[-i{{e}^{-{{e}^{-iz}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer52)
Let R and S be two non-void relations in a set A. which of the following statements is not true?
A)
R and S transitive\[\Rightarrow \] \[R\cup S\] is transitive done
clear
B)
R and S transitive\[\Rightarrow R\cap S\] is transitive done
clear
C)
R and S transitive\[\Rightarrow R\cup S\] is symmetric done
clear
D)
R and S reflexive\[\Rightarrow R\cap S\] is reflexive done
clear
View Solution play_arrow
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question_answer53)
Let function\[f:R\to R\] be defined by \[f(x)=2x+sinx\] for \[x\in R,\] then f is
A)
One-one and onto done
clear
B)
one-one but NOT onto done
clear
C)
onto but NOT one-one done
clear
D)
Neither one-one nor onto done
clear
View Solution play_arrow
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question_answer54)
If \[f(x)=5lo{{g}_{5}}x\] then \[{{f}^{-1}}(\alpha -\beta )\] where \[\alpha ,\beta \in R\]is equal to
A)
\[{{f}^{-1}}(\alpha )-{{f}^{-1}}(\beta )\] done
clear
B)
\[\frac{{{f}^{-1}}(\alpha )}{{{f}^{-1}}(\beta )}\] done
clear
C)
\[\frac{1}{f(\alpha -\beta )}\] done
clear
D)
\[\frac{1}{f(\alpha )-f(\beta )}\] done
clear
View Solution play_arrow
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question_answer55)
If \[f(x)=\left| x-2 \right|\] and \[g(x)=f[f(x)],\] Then for \[x>20,g'(x)\] is equal to
A)
\[-1\] done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer56)
Let [x] denote the greatest integer \[\le x.\] If \[f(x)=[x]\] and \[g(x)=\left| x \right|,\]then the value of \[f\left( g\left( \frac{8}{5} \right) \right)-g\left( f\left( -\frac{8}{5} \right) \right)\]is
A)
2 done
clear
B)
-2 done
clear
C)
1 done
clear
D)
-1 done
clear
View Solution play_arrow
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question_answer57)
If f(x) is an invertible function and \[g(x)=2f(x)+5,\] then the value of \[{{g}^{-1}}(x)\] is
A)
\[2{{f}^{-1}}(x)-5\] done
clear
B)
\[\frac{1}{2{{f}^{-1}}(x)+5}\] done
clear
C)
\[\frac{1}{2}{{f}^{-1}}(x)+5\] done
clear
D)
\[{{f}^{-1}}\left( \frac{x-5}{2} \right)\] done
clear
View Solution play_arrow
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question_answer58)
Let S be any set and P (S) be its power set, We define a relation R on P(S) by ARB to mean \[A\subseteq B;\forall A,B\in P(S).\] Then R is
A)
Equivalence relation done
clear
B)
Not an equivalence but partial order relation done
clear
C)
Both equivalence and partial order relation done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer59)
Let \[f:\{2,3,4,5\}\to \{3,4,5,9\}\] and \[g:\{3,4,5,9\}\]\[\to \{7,11,15\}\] be functions defined as \[f(2)=3f(3)=4,f(4)=f(5)=5,g(3)=g(4)=7,\] and \[g(5)=g(9)=11.\] Then gof (5) is equal to
A)
5 done
clear
B)
7 done
clear
C)
11 done
clear
D)
1 done
clear
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question_answer60)
If \[f(x)=\sqrt{3\left| x \right|-x-2}\] and \[g(x)=sinx,\] then domain of definition of fog (x) is
A)
\[\left\{ 2n\pi +\frac{\pi }{2} \right\},n\in I\] done
clear
B)
\[\underset{n\in I}{\mathop{\bigcup }}\,\]\[\left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right)\] done
clear
C)
\[\left( 2n\pi +\frac{7\pi }{6} \right),n\in I\] done
clear
D)
\[\{(4m+1)\frac{\pi }{2}:m\in I\}\underset{n\in I}{\mathop{\bigcup }}\,\left[ \left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right) \right]\] done
clear
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question_answer61)
If \[f:R\to R,g:R\to R\] and \[h:R\to R\]are such that \[f(x)={{x}^{2}},g(x)=tanx\]and \[h(x)=logx,\]then the value of \[(ho(gof))(x)ifx=\sqrt{\frac{\pi }{4}}\] will be
A)
0 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
\[\pi \] done
clear
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question_answer62)
If \[f(x)=\left\{ \begin{matrix} {{x}^{3}}+1,x<0 \\ {{x}^{2}}+1,x\ge 0 \\ \end{matrix},g(x)=\left\{ \begin{matrix} {{(x-1)}^{1/3}},x<1 \\ {{(x-1)}^{1/3}},x\ge 1 \\ \end{matrix}, \right. \right.\] Then (gof) (x) is equal to
A)
\[x,\forall x\in R\] done
clear
B)
\[x-1,\forall x\in R\] done
clear
C)
\[x+1,\forall x\in R\] done
clear
D)
None of these done
clear
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question_answer63)
If \[f(x)=2x+\left| x \right|,g(x)=\frac{1}{3}(2x-\left| x \right|)\] and \[h(x)=f(g(x)),\] then domain of \[{{\sin }^{-1}}\]\[\underbrace{(h(h(h(h...h(x)...))))}_{n\,times}\] is
A)
\[[-1,1]\] done
clear
B)
\[\left[ -1,-\frac{1}{2} \right]\cup \left[ \frac{1}{2},1 \right]\] done
clear
C)
\[\left[ -1,-\frac{1}{2} \right]\] done
clear
D)
\[\left[ \frac{1}{2},1 \right]\] done
clear
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question_answer64)
Let \[A=Z\cup \{\sqrt{2}\}.\] Define a relation R in A by aRb if and only if \[a+b\in Z.\] The relation R is
A)
Reflexive done
clear
B)
Symmetric and transitive done
clear
C)
Only transitive done
clear
D)
None of these done
clear
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question_answer65)
If \[f(x)=\frac{ax+d}{cx+b}\] and \[f[f(x)]=x\] for all x, then:
A)
\[a=b\] done
clear
B)
\[c=d\] done
clear
C)
\[a+b=0\] done
clear
D)
\[c+d=0\] done
clear
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question_answer66)
If \[f(x)=si{{n}^{2}}x+{{\sin }^{2}}\left( x+\frac{\pi }{3} \right)+\cos x\cos \left( x+\frac{\pi }{3} \right)\]and \[g\left( \frac{5}{4} \right)=1,\] then gof(x)=
A)
1 done
clear
B)
0 done
clear
C)
\[\sin x\] done
clear
D)
None done
clear
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question_answer67)
For real numbers x and y, we define x R y iff \[x-y+\sqrt{5}\] is an irrational number. The relation R is
A)
Reflexive done
clear
B)
Symmetric done
clear
C)
Transitive done
clear
D)
None of these done
clear
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question_answer68)
Let \[f(x)=-1+\left| x-1 \right|,-1\le x\le 3\] and \[\le g(x)=2-\left| x+1 \right|,-2\le x\le 2,\] then \[(fog)(x)\], is equal to
A)
\[\left\{ \begin{matrix} x+1-2\le x\le 0 \\ x-10<x\le 2 \\ \end{matrix} \right.\] done
clear
B)
\[\left\{ \begin{matrix} x-1-2\le x\le 0 \\ x+10<x\le 2 \\ \end{matrix} \right.\] done
clear
C)
\[\left\{ \begin{matrix} -1-x-2\le x\le 0 \\ x-10<x\le 2 \\ \end{matrix} \right.\] done
clear
D)
None of these done
clear
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question_answer69)
Let \[f:R\to R\] be a function such that \[f(x)=ax+3sinx+4cosx.\] Then f(x) is invertible if
A)
\[a\in (-5,5)\] done
clear
B)
\[a\in (-\infty ,-5)\] done
clear
C)
\[a\in (0,+\infty )\] done
clear
D)
None of these done
clear
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question_answer70)
Let \[f:(4,6)\to (6,8)\] be a function defined by \[f(x)=x+\left[ \frac{x}{2} \right]\] (where [.] denotes the greatest integer function), then \[{{f}^{-1}}(x)\] is equal to
A)
\[x-\left[ \frac{x}{2} \right]\] done
clear
B)
\[-x-2\] done
clear
C)
\[x-2\] done
clear
D)
\[\frac{1}{x+\left[ \frac{x}{2} \right]}\] done
clear
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